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An Introduction to Vectors and Tensors from a Computational Perspective

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An Introduction to Vectors and Tensors from a Computational Perspective An Introduction to Vectors and Tensors from a Computational Perspective UTC-CECS-SimCenter-2014-01 August 2014 GRADUATE SCHOOL OF COMPUTATIONAL ENGINEERING 701 East M.L. King Boulevard • Chattanooga, TN 37403 AN INTRODUCTION TO VECTORS AND TENSORS FROM A COMPUTATIONAL PERSPECTIVE W. Roger Briley SimCenter: National Center for Computational Engineering University of Tennessee at Chattanooga August 2014 This report is an updated version of Report UTC-CECS-SimCenter-2012-01. Preface This report is intended to provide a self-contained introduction to Cartesian tensors for students just entering graduate school in engineering and science majors, especially those interested in computational engineering and applied computational science. This introduction assumes students have a background in multivariable calculus but no familiarity with tensors. _____________________________________________________________________________________________________________________ TABLE OF CONTENTS 1. Introduction ............................................................................................................................. 2 2. Vector Basics ........................................................................................................................... 4 3. Index Notation for Vectors, Tensors and Matrices ..................................................................4 4. Tensor Basics .......................................................................................................................... 7 5. Vector and Tensor Fields .........................................................................................................9 6. Calculus Operations in Cartesian Tensor Notation ..................................................................9 7. Transformation Laws for Cartesian Coordinates and Tensor Components ...........................11 8. Transformation from Cartesian to General Curvilinear Coordinates ....................................14 REFERENCES ......................................................................................................................15 APPENDIX – Dual-Basis Vector Calculus ...........................................................................16 _____________________________________________________________________________________________________________________ © Copyright: SimCenter: National Center for Computational Engineering, 2012-2014 SimCenter: National Center for Computational Engineering • UT Chattanooga • 8/23/2014 1 AN INTRODUCTION TO VECTORS AND TENSORS FROM A COMPUTATIONAL PERSPECTIVE _____________________________________________________________________________________________________________________ 1. Introduction 1.1 Vectors and Tensors in Physics Magnitude and direction are common geometric properties of physical entities. Scalars are physical quantities such as density and temperature that have magnitude (measured in a specified system of units) but no directional orientation. Vectors are physical quantities such as velocity and force with magnitude (length) and a single direction. The direction of vectors can be defined only in relation to a specified set of N reference directions that comprise a frame of reference for the N-dimensional physical space considered: typically N = 1, 2 or 3. The reference frame could be a set of unit vectors or a coordinate system. Vector magnitude and direction are quantified by N scalar components that are defined by scalar projection onto these directions. Although vector components depend on the choice of reference directions, the magnitude and direction of the vector are invariant physical properties that are independent of the frame of reference. Tensors are physical quantities such as stress and strain that have magnitude and two or more directions. For example, stress is a relationship between force and area (magnitude and two directions) and is thus a second-order tensor with N 2 components. Tensors also have invariant physical properties that are coordinate independent. True physical tensors of order higher than two are uncommon, but higher order tensors are common in mathematical descriptions of physics. 1.2 Vectors and Tensors in Mathematics Mathematically, vectors and tensors describe physical entities and their mathematical abstractions as directional objects represented by scalar components that are defined by projection onto a specified set of base vectors (possibly unit vectors) comprising a basis, and that satisfy transformation laws for a change of basis. It is important to recognize that the term tensor is a general mathematical description for geometric objects that have magnitude and any number of directions. A tensor of order p has content from p directions and has N p components. Thus a scalar is a zeroth-order tensor, a vector is a first-order tensor, and so on. 1.3 A Computational Perspective The present introduction will consider vectors and tensors as encountered in computational simulations of physical fields in which scalar, vector and tensor quantities vary with position in space and with time. Fields require a coordinate system to locate points in space. Vector and tensor fields also require a local basis at each point to define vector/tensor components. Disconflation of Vector Bases and Coordinates Systems - Most mathematical treatments of tensors assume that the local basis is aligned with the coordinate directions: cf. [1,2] . However, the alignment of base vectors and coordinate directions introduces complexity in curvilinear SimCenter: National Center for Computational Engineering • UT Chattanooga • 8/23/2014 2 orthogonal and especially in nonorthogonal coordinates that is not present in Cartesian coordinates: The local basis is constant for Cartesian coordinates but varies with spatial position using any curvilinear coordinate system. Nonorthogonal coordinates introduce a dual basis: one basis is parallel to the coordinate lines (the contravariant basis) and a reciprocal basis (the covariant basis) is perpendicular to coordinate tangent planes. These parallel and reciprocal bases coincide for curvilinear orthogonal coordinates but vary with spatial position. The resulting complications include differentiation of spatially varying base vectors, the metric tensor, dual base vectors, contravariant and covariant components, tensor differentiation, and Christoffel symbols. The calculus of tensors in general nonorthogonal coordinates is therefore significantly more complicated than that of Cartesian tensors. However, the complexity of variable-direction and nonorthogonal base vectors in general coordinates is commonly avoided in computational solution of both differential and integral conservation laws in discrete form, whether using structured or unstructured grids. Structured Grids - In differential approaches using coordinate systems, the technique used is to transform spatial derivative terms from Cartesian to general curvilinear coordinates, while retaining a uniform Cartesian local basis for vector/tensor components. The governing equations are thereby written in general curvilinear coordinates, but the Cartesian vector/tensor components remain as dependent variables. This approach is widely discussed and has been standard practice in computational fluid dynamics for many years: cf. [3,4,5]. The use of a uniform Cartesian local basis may also reduce spatial discretization error in computations. The reason is that spatial variation in base vectors causes extraneous, nonphysical spatial variation in components that when differentiated may require higher local resolution than Cartesian components: for example in regions of large coordinate curvature. Unstructured Grids - Another common approach for spatial discretization uses unstructured grids, for which no identifiable family of coordinate lines exists: cf. [6,7]. The governing equations are typically written in integral rather than differential form, with discrete integral approximations that require multidimensional interpolation of vectors and tensors rather than differentiation. A pointwise Cartesian local basis is used for vector/tensor components, with a local rotation of Cartesian unit vectors for alignment with local surfaces as needed in the interpolation process. Finally, it should be emphasized that tensor mathematics is a broad area of study that can be far more complicated than what is needed and discussed here as background for computational field simulation. The present introduction covers basic material that is fundamental to the understanding and computation of physical vector and tensor fields. It is hoped that the present effort to disconflate the local vector basis and coordinate system will provide useful insight into the computation of tensor fields. The main body of this report addresses Cartesian coordinates and basis vectors and that are not necessarily aligned. For those who are interested, the APPENDIX gives a summary of dual-basis vector calculus for general curvilinear coordinates. Detailed discussions of vectors and tensors are given in [1,2,8] and in many other references. SimCenter: National Center for Computational Engineering • UT Chattanooga • 8/23/2014 3 2. Vector Basics First consider a vector a with base O and tip A, as shown in the sketch. The vector is a directed line segment (arrow) that has inherent magnitude and direction. The vector is called a free vector if its location is not specified and a fixed or bound vector if the base O has a specific location in space. The direction
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